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<h1 class="mume-header" id="%E7%AC%AC%E4%B8%80%E9%83%A8%E5%88%86%E8%B4%9D%E5%8F%B6%E6%96%AF%E7%BD%91%E5%9F%BA%E7%A1%80">&#x7B2C;&#x4E00;&#x90E8;&#x5206;&#xFF1A;&#x8D1D;&#x53F6;&#x65AF;&#x7F51;&#x57FA;&#x7840;</h1>

</section><section lineno="28" class="slide " data-line="28" data-h="2" data-v="0">
<h2 class="mume-header" id="11-%E4%BF%A1%E6%81%AF%E8%AE%BA%E5%9F%BA%E7%A1%80">1.1 &#x4FE1;&#x606F;&#x8BBA;&#x57FA;&#x7840;</h2>

<p>&#x4FE1;&#x606F;&#x8BBA;&#x662F;&#x57FA;&#x4E8E;&#x6982;&#x7387;&#x8BBA;&#x7684;&#x4E00;&#x95E8;&#x7814;&#x7A76;&#x4FE1;&#x606F;&#x4F20;&#x8F93;&#x548C;&#x5904;&#x7406;&#x7684;&#x6570;&#x5B66;&#x7406;&#x8BBA;&#x3002;&#x5B83;&#x4E0D;&#x4EC5;&#x662F;&#x4FE1;&#x606F;&#x6280;&#x672F;&#x7684;&#x57FA;&#x7840;&#xFF0C;&#x4E5F;&#x5728;&#x7EDF;&#x8BA1;&#x529B;&#x5B66;&#x3001;&#x673A;&#x5668;&#x5B66;&#x4E60;&#x7B49;&#x9886;&#x57DF;&#x53D1;&#x6325;&#x91CD;&#x8981;&#x4F5C;&#x7528;&#x3002;&#x5728;&#x6784;&#x5EFA;&#x8D1D;&#x53F6;&#x65AF;&#x7F51;&#x7684;&#x8FC7;&#x7A0B;&#x4E2D;&#xFF0C;&#x53EF;&#x4EE5;&#x7528;&#x4FE1;&#x606F;&#x8BBA;&#x6765;&#x8FDB;&#x884C;&#x5206;&#x6790;&#x3002;</p>
</section><section><section lineno="32" class="slide " data-line="32" data-h="3" data-v="0">
<h3 class="mume-header" id="111-%E9%A2%84%E5%A4%87%E6%95%B0%E5%AD%A6%E7%9F%A5%E8%AF%86jensen%E4%B8%8D%E7%AD%89%E5%BC%8F">1.1.1 &#x9884;&#x5907;&#x6570;&#x5B66;&#x77E5;&#x8BC6;&#xFF1A;Jensen&#x4E0D;&#x7B49;&#x5F0F;</h3>

<p>Jesen&#x4E0D;&#x7B49;&#x5F0F;&#x6E90;&#x4E8E;&#x51FD;&#x6570;&#x7684;&#x51F9;&#x51F8;&#x6027;&#x3002;&#x5728;&#x6570;&#x5B66;&#x4E2D;&#xFF0C;&#x79F0;&#x4E00;&#x4E2A;&#x51FD;&#x6570;&#x4E3A;&#x51F9;&#x51FD;&#x6570;&#x662F;&#x6307;&#x5411;&#x4E0A;&#x51F9;&#xFF0C;&#x51F8;&#x51FD;&#x6570;&#x662F;&#x6307;&#x5411;&#x4E0B;&#x51F8;&#xFF0C;&#x5982;&#x4E0B;&#x56FE;&#x6240;&#x793A;&#x3002;<br>
<img src="https://img-blog.csdnimg.cn/img_convert/813b66b360efdda30288467371e8a8ce.png#pic_center" alt="&#x51F9;&#x51FD;&#x6570;&#x548C;&#x51F8;&#x51FD;&#x6570;.png"></p>
</section><section vertical="true" lineno="36" class="slide " data-line="36" data-h="3" data-v="1">
<p>&#x82E5;&#x4E00;&#x4E2A;&#x51FD;&#x6570;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span>&#x5728;&#x5B9E;&#x6570;&#x8F74;&#x7684;&#x67D0;&#x4E2A;&#x533A;&#x95F4;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span>&#x4E0A;&#x88AB;&#x79F0;&#x4E3A;&#x51F9;&#x51FD;&#x6570;&#xFF0C;&#x5219;&#x6709;&#x5982;&#x4E0B;&#x4E0D;&#x7B49;&#x5F0F;&#x6210;&#x7ACB;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">&#x2200;</mi><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>&#x2208;</mo><mi>I</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>&#x3BB;</mi><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x3BB;</mi><mo stretchy="false">)</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>&#x2265;</mo><mi>&#x3BB;</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x3BB;</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mi mathvariant="normal">&#x2200;</mi><mi>&#x3BB;</mi><mo>&#x2208;</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\forall x_1,x_2\in I: f(\lambda x_1+(1-\lambda)x_2)\geq \lambda f(x_1)+(1-\lambda)f(x_2),\forall\lambda\in [0,1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">&#x2200;</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2208;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">&#x3BB;</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">&#x3BB;</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">&#x3BB;</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">&#x3BB;</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">&#x2200;</span><span class="mord mathnormal">&#x3BB;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2208;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span></span><br>
&#x82E5;&#x5F0F;&#x4E2D;&#x7B49;&#x53F7;&#x53EA;&#x5728;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1=x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x65F6;&#x6210;&#x7ACB;&#xFF0C;&#x5219;&#x79F0;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span>&#x5728;&#x533A;&#x95F4;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span>&#x4E0A;&#x4E25;&#x683C;&#x51F9;&#x3002;&#x82E5;&#x5C06;&#x5F0F;&#x4E2D;&#x4E0D;&#x7B49;&#x53F7;&#x65B9;&#x5411;&#x6539;&#x53D8;&#xFF0C;&#x5219;&#x5F97;&#x5230;&#x51F8;&#x51FD;&#x6570;&#x7684;&#x5B9A;&#x4E49;&#x3002;<br>
&#x6211;&#x4EEC;&#x770B;&#x5230;&#xFF0C;&#x5F0F;&#x4E2D;&#x53C2;&#x6570;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>&#x3BB;</mi><mo separator="true">,</mo><mn>1</mn><mo>&#x2212;</mo><mi>&#x3BB;</mi></mrow><annotation encoding="application/x-tex">\lambda,1-\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">&#x3BB;</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">&#x3BB;</span></span></span></span>&#x53EF;&#x4EE5;&#x770B;&#x505A;&#x6982;&#x7387;&#xFF0C;&#x7531;&#x5B83;&#x4EEC;&#x4E4B;&#x548C;&#x4E3A;1&#xFF0C;&#x4E0D;&#x7B49;&#x53F7;&#x5DE6;&#x4FA7;&#x53EF;&#x4EE5;&#x770B;&#x4F5C;&#x8F93;&#x5165;&#x7684;&#x671F;&#x671B;&#x5BF9;&#x5E94;&#x7684;&#x8F93;&#x51FA;&#xFF0C;&#x53F3;&#x4FA7;&#x53EF;&#x4EE5;&#x770B;&#x4F5C;&#x8F93;&#x5165;&#x5BF9;&#x5E94;&#x7684;&#x8F93;&#x51FA;&#x7684;&#x671F;&#x671B;&#x3002;&#x5BF9;&#x4E8E;&#x4E0A;&#x51F9;&#x51FD;&#x6570;&#xFF0C;&#x8F93;&#x5165;&#x7684;&#x671F;&#x671B;&#x5BF9;&#x5E94;&#x7684;&#x8F93;&#x51FA;&#x5927;&#x4E8E;&#x8F93;&#x5165;&#x5BF9;&#x5E94;&#x7684;&#x8F93;&#x51FA;&#x7684;&#x671F;&#x671B;&#xFF1B;&#x5BF9;&#x4E8E;&#x4E0B;&#x51F8;&#x51FD;&#x6570;&#xFF0C;&#x5219;&#x521A;&#x597D;&#x76F8;&#x53CD;&#x3002;<br>
Jensen&#x4E0D;&#x7B49;&#x5F0F;&#x5219;&#x662F;&#x4E0A;&#x5F0F;&#x5728;&#x671F;&#x671B;&#x610F;&#x4E49;&#x4E0A;&#x7684;&#x63A8;&#x5E7F;&#x3002;</p>
</section><section vertical="true" lineno="44" class="slide " data-line="44" data-h="3" data-v="2">
<p><strong>Jensen&#x4E0D;&#x7B49;&#x5F0F;</strong><br>
&#x8BBE;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span>&#x4E3A;&#x533A;&#x95F4;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span>&#x4E0A;&#x7684;&#x51F9;&#x51FD;&#x6570;&#xFF0C;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>&#x2208;</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>&#x22EF;</mo><mtext>&#x2009;</mtext><mo separator="true">,</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">p_i\in [0,1],i=1,2,\cdots,n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2208;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">&#x22EF;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span></span></span></span>&#xFF0C;&#x4E14;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>p</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sum_{i=1}^n p_i=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.104002em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">&#x2211;</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>&#xFF0C;&#x5219;&#x5BF9;&#x4EFB;&#x4F55;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>&#x2208;</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x_i\in I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2208;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span></span>&#xFF0C;&#x6709;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>f</mi><mo stretchy="false">(</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>p</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>&#x2265;</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>p</mi><mi>i</mi></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(1)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">f(\sum_{i=1}^n p_i x_i)\geq \sum_{i=1}^n p_i f(x_i) \tag{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span></span><span class="mord">)</span></span></span></span></span></span><br>
&#x82E5;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span>&#x4E25;&#x683C;&#x51F9;&#xFF0C;&#x5219;&#x4E0A;&#x5F0F;&#x7B49;&#x53F7;&#x53EA;&#x5728;&#x4E0B;&#x5217;&#x6761;&#x4EF6;&#x6EE1;&#x8DB3;&#x65F6;&#x624D;&#x6210;&#x7ACB;&#xFF1A;&#x82E5;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>&#x22C5;</mo><msub><mi>p</mi><mi>j</mi></msub><mo mathvariant="normal">&#x2260;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p_i\cdot p_j\neq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63889em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x22C5;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mord"><span class="mrel">&#xE020;</span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>&#xFF0C;&#x5219;&#x5FC5;&#x6709;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><msub><mi>x</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">x_i=x_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>&#x3002;</p>
</section><section vertical="true" lineno="51" class="slide " data-line="51" data-h="3" data-v="3">
<p><strong>&#x8BC1;&#x660E;</strong><br>
&#x7528;&#x5F52;&#x7EB3;&#x6CD5;&#x8BC1;&#x660E;&#xFF0C;&#x5F53;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>&#x65F6;&#xFF0C;&#x5F0F;(1)&#x6052;&#x7B49;&#x3002;&#x5047;&#x8BBE;&#x5F0F;(1)&#x5728;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n=k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>&#x65F6;&#x6210;&#x7ACB;&#xFF0C;&#x8BC1;&#x660E;&#x5B83;&#x5728;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=k+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>&#x65F6;&#x4E5F;&#x6210;&#x7ACB;&#xFF0C;&#x5373;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>f</mi><mo stretchy="false">(</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></munderover><msub><mi>p</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><msub><mi>p</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>f</mi><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>&#x2212;</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mfrac><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><msub><mi>p</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo fence="true">]</mo></mrow><mo stretchy="false">(</mo><mtext>&#x5047;&#x8BBE;</mtext><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo mathvariant="normal">&#x2260;</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>&#x2265;</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>f</mi><mrow><mo fence="true">(</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>&#x2212;</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mfrac><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><msub><mi>p</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mtext>&#x6839;&#x636E;&#x51F9;&#x51FD;&#x6570;&#x5B9A;&#x4E49;</mtext><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>f</mi><mrow><mo fence="true">(</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mfrac><msub><mi>p</mi><mi>i</mi></msub><mrow><mn>1</mn><mo>&#x2212;</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mfrac><msub><mi>x</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>&#x2265;</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mfrac><msub><mi>p</mi><mi>i</mi></msub><mrow><mn>1</mn><mo>&#x2212;</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mfrac><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mtext>&#x6839;&#x636E;&#x5F52;&#x7EB3;&#x5047;&#x8BBE;</mtext><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><msub><mi>p</mi><mi>i</mi></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>p</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></munderover><msub><mi>p</mi><mi>i</mi></msub><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
f(\sum_{i=1}^{k+1} p_i x_i) &amp;= f(\sum_{i=1}^k p_i x_i + p_{k+1} x_{k+1}) \\
&amp;= f \left[(1-p_{k+1})\frac{1}{1-p_{k+1}}\sum_{i=1}^k p_i x_i + p_{k+1} x_{k+1}\right] (&#x5047;&#x8BBE;p_{k+1}\neq 1) \\
&amp;\geq (1-p_{k+1})f \left(\frac{1}{1-p_{k+1}}\sum_{i=1}^k p_i x_i\right)+p_{k+1}f(x_{k+1}) (&#x6839;&#x636E;&#x51F9;&#x51FD;&#x6570;&#x5B9A;&#x4E49;) \\
&amp;= (1-p_{k+1})f \left(\sum_{i=1}^k \frac{p_i}{1-p_{k+1}} x_i\right)+p_{k+1}f(x_{k+1}) \\
&amp;\geq (1-p_{k+1})\sum_{i=1}^k \frac{p_i}{1-p_{k+1}}f(x_i) +p_{k+1}f(x_{k+1}) (&#x6839;&#x636E;&#x5F52;&#x7EB3;&#x5047;&#x8BBE;) \\
&amp;=\sum_{i=1}^k p_i f(x_i) +p_{k+1} f(x_k+1) \\
&amp;=\sum_{i=1}^{k+1} p_i f(x_i)
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:23.896474000000005em;vertical-align:-11.698237000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:12.198237000000002em;"><span style="top:-14.198237000000002em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361130000000003em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-10.784455000000003em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"></span></span><span style="top:-7.370673000000002em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"></span></span><span style="top:-3.9568910000000006em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"></span></span><span style="top:-0.5431089999999992em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"></span></span><span style="top:2.870673000000001em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"></span></span><span style="top:6.284455000000002em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:11.698237000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:12.198237000000002em;"><span style="top:-14.198237000000002em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361130000000003em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-10.784455000000003em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">[</span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.3139999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.894331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361130000000003em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord cjk_fallback">&#x5047;&#x8BBE;</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mord"><span class="mrel">&#xE020;</span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-7.370673000000002em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.3139999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.894331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361130000000003em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord cjk_fallback">&#x6839;&#x636E;&#x51F9;&#x51FD;&#x6570;&#x5B9A;&#x4E49;</span><span class="mclose">)</span></span></span><span style="top:-3.9568910000000006em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361130000000003em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.3139999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.894331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-0.5431089999999992em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361130000000003em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.3139999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.894331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord cjk_fallback">&#x6839;&#x636E;&#x5F52;&#x7EB3;&#x5047;&#x8BBE;</span><span class="mclose">)</span></span></span><span style="top:2.870673000000001em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361130000000003em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:6.284455000000002em;"><span class="pstrut" style="height:3.836113em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8361130000000003em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:11.698237000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span><br>
&#x547D;&#x9898;&#x5F97;&#x8BC1;&#x3002;</p>
</section><section vertical="true" lineno="66" class="slide " data-line="66" data-h="3" data-v="4">
<p>Jensen&#x4E0D;&#x7B49;&#x5F0F;&#x662F;&#x4E0E;&#x51FD;&#x6570;&#x51F9;&#x51F8;&#x6027;&#x6709;&#x5173;&#x7684;&#x57FA;&#x672C;&#x6027;&#x8D28;&#xFF0C;&#x5728;&#x4FE1;&#x606F;&#x8BBA;&#x4E2D;&#x5E38;&#x4F1A;&#x7528;&#x5230;&#xFF0C;&#x6BD4;&#x5982;&#x7528;&#x4E8E;&#x8BA1;&#x7B97;&#x4FE1;&#x606F;&#x71B5;&#x7684;&#x5BF9;&#x6570;&#x51FD;&#x6570;&#x5C31;&#x6EE1;&#x8DB3;&#x51F9;&#x51FD;&#x6570;&#x7684;Jensen&#x4E0D;&#x7B49;&#x5F0F;&#xFF0C;&#x8FD9;&#x5728;&#x540E;&#x6587;&#x8BC1;&#x660E;&#x4FE1;&#x606F;&#x71B5;&#x7684;&#x6027;&#x8D28;&#x65F6;&#x4F1A;&#x7528;&#x5230;&#x3002;</p>
</section></section><section><section lineno="69" class="slide " data-line="69" data-h="4" data-v="0">
<h3 class="mume-header" id="112-%E7%86%B5">1.1.2 &#x71B5;</h3>

<p>&#x4E00;&#x4E2A;&#x79BB;&#x6563;&#x968F;&#x673A;&#x53D8;&#x91CF;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>&#x7684;&#x71B5;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span>&#x7684;&#x5B9A;&#x4E49;&#x4E3A;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mi>P</mi></msub><mo stretchy="false">[</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">]</mo><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(X)=E_P[\log{\frac{1}{P(X)}}]=-\sum_X P(X)\log{P(X)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">[</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.344341em;vertical-align:-1.294336em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></span></span><br>
&#x5176;&#x4E2D;&#xFF0C;&#x7EA6;&#x5B9A;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>1</mn><mn>0</mn></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0\log{\frac{1}{0}}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>.&#x4E0A;&#x5F0F;&#x7684;&#x5BF9;&#x6570;&#x82E5;&#x4EE5;2&#x4E3A;&#x5E95;&#xFF0C;&#x5219;&#x71B5;&#x7684;&#x5355;&#x4F4D;&#x662F;&#x6BD4;&#x7279;&#xFF0C;&#x82E5;&#x4EE5;e&#x4E3A;&#x5E95;&#xFF0C;&#x5219;&#x5355;&#x4F4D;&#x662F;&#x5948;&#x7279;&#xFF0C;&#x540E;&#x6587;&#x5C06;&#x90FD;&#x4EE5;&#x6BD4;&#x7279;&#x4E3A;&#x5355;&#x4F4D;&#x3002;</p>
</section><section vertical="true" lineno="76" class="slide " data-line="76" data-h="4" data-v="1">
<p>&#x71B5;&#x5728;&#x70ED;&#x529B;&#x5B66;&#x4E2D;&#x8868;&#x793A;&#x7CFB;&#x7EDF;&#x7684;&#x6DF7;&#x4E71;&#x7A0B;&#x5EA6;&#xFF0C;&#x5728;&#x6982;&#x7387;&#x8BBA;&#x4E2D;&#x8868;&#x793A;&#x968F;&#x673A;&#x53D8;&#x91CF;&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x7A0B;&#x5EA6;&#xFF0C;&#x5728;&#x4FE1;&#x606F;&#x8BBA;&#x4E2D;&#x8868;&#x793A;&#x5BF9;&#x4FE1;&#x6E90;&#x7684;&#x671F;&#x671B;&#x7F16;&#x7801;&#x957F;&#x5EA6;&#x3002;<br>
&#x5148;&#x6765;&#x89E3;&#x91CA;&#x4E0B;&#x4FE1;&#x606F;&#x8BBA;&#x4E2D;&#x671F;&#x671B;&#x7F16;&#x7801;&#x957F;&#x5EA6;&#xFF1A;&#x5047;&#x8BBE;&#x6709;&#x4E00;&#x4E2A;&#x4FE1;&#x6E90;&#xFF0C;&#x53EF;&#x4EA7;&#x751F;A&#x3001;B&#x3001;C&#x4E09;&#x79CD;&#x4E0D;&#x540C;&#x7684;&#x4FE1;&#x606F;&#xFF0C;&#x4EA7;&#x751F;&#x7684;&#x6982;&#x7387;&#x5206;&#x522B;&#x4E3A;1/2&#x3001;1/4&#x548C;1/4&#xFF0C;&#x6211;&#x4EEC;&#x8981;&#x8BBE;&#x8BA1;&#x4E00;&#x5957;&#x7F16;&#x7801;&#x7CFB;&#x7EDF;&#x6765;&#x8BB0;&#x5F55;&#x8FD9;&#x4E2A;&#x4FE1;&#x6E90;&#x6240;&#x4EA7;&#x751F;&#x7684;&#x4FE1;&#x606F;&#xFF0C;&#x6240;&#x7528;&#x7684;&#x6BD4;&#x7279;&#x4F4D;&#x6570;&#x8D8A;&#x5C11;&#x8D8A;&#x597D;&#x3002;&#x663E;&#x7136;&#xFF0C;&#x6211;&#x4EEC;&#x5E94;&#x8BE5;&#x4E3A;&#x51FA;&#x73B0;&#x6982;&#x7387;&#x5927;&#x7684;&#x4FE1;&#x606F;&#x5206;&#x914D;&#x7801;&#x957F;&#x8F83;&#x77ED;&#x7684;&#x7F16;&#x7801;&#xFF0C;&#x5176;&#x957F;&#x5EA6;&#x53EF;&#x901A;&#x8FC7;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\log{\frac{1}{P(X)}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.365108em;vertical-align:-0.52em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>&#x6765;&#x786E;&#x5B9A;&#xFF0C;&#x6BD4;&#x5982;&#x6211;&#x4EEC;&#x4E3A;A&#x5206;&#x914D;&#x7801;&#x957F;&#x4E3A;1&#x7684;&#x7F16;&#x7801;&#xFF0C;&#x4E3A;B&#x548C;C&#x5206;&#x914D;&#x7801;&#x957F;&#x4E3A;2&#x7684;&#x7F16;&#x7801;&#xFF0C;&#x901A;&#x8FC7;&#x970D;&#x592B;&#x66FC;&#x7F16;&#x7801;&#x7B97;&#x6CD5;&#xFF0C;&#x53EF;&#x5C06;A&#x7F16;&#x7801;&#x4E3A;0&#xFF0C;&#x5C06;B&#x548C;C&#x7F16;&#x7801;&#x4E3A;10&#x548C;11.&#x6B64;&#x65F6;&#xFF0C;&#x8BE5;&#x4FE1;&#x6E90;&#x7684;&#x7F16;&#x7801;&#x5E73;&#x5747;&#x7801;&#x957F;&#x5219;&#x4E3A;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>&#xD7;</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>&#xD7;</mo><mn>2</mn><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>&#xD7;</mo><mn>2</mn><mo>=</mo><mn>1.5</mn></mrow><annotation encoding="application/x-tex">\frac{1}{2}\times 1+\frac{1}{4}\times 2+\frac{1}{4}\times 2=1.5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#xD7;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#xD7;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#xD7;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1.5</span></span></span></span></p>
</section><section vertical="true" lineno="79" class="slide " data-line="79" data-h="4" data-v="2">
<p><strong>&#x970D;&#x592B;&#x66FC;&#x7F16;&#x7801;&#x7B97;&#x6CD5;&#xFF1A;</strong><br>
Huffman&#x4F7F;&#x7528;&#x81EA;&#x5E95;&#x5411;&#x4E0A;&#x6784;&#x5EFA;&#x4E8C;&#x53C9;&#x6811;&#x7684;&#x65B9;&#x5F0F;&#xFF0C;&#x6784;&#x5EFA;&#x7F16;&#x7801;&#x6811;&#xFF0C;&#x6BCF;&#x4E2A;&#x5B57;&#x7B26;&#x7684;&#x6700;&#x7EC8;&#x7F16;&#x7801;&#x662F;&#x4ECE;&#x6839;&#x8D70;&#x5230;&#x53F6;&#x5B50;&#x7684;0/1&#x5E8F;&#x5217;&#xFF0C;&#x5F80;&#x5DE6;&#x8D70;&#x4E3A;0&#xFF0C;&#x5F80;&#x53F3;&#x8D70;&#x4E3A;1&#x3002;<br>
&#x6784;&#x5EFA;&#x7684;&#x65B9;&#x5F0F;&#x4E5F;&#x5F88;&#x7B80;&#x5355;&#xFF1A;<br>
a. &#x521D;&#x59CB;&#x96C6;&#x5408;&#x91CC;&#x6709;n&#x68F5;&#x6811;&#xFF0C;&#x5206;&#x522B;&#x8868;&#x793A;&#x5F85;&#x7F16;&#x7801;&#x7684;n&#x4E2A;&#x5B57;&#x7B26;&#xFF0C;&#x6811;&#x6839;&#x7684;&#x503C;&#x662F;&#x8FD9;&#x4E2A;&#x5B57;&#x7B26;&#x51FA;&#x73B0;&#x7684;&#x6982;&#x7387;&#x3002;<br>
b. &#x9009;&#x62E9;&#x6811;&#x6839;&#x503C;&#x6700;&#x5C0F;&#x7684;&#x4E24;&#x9897;&#x6811;&#xFF0C;&#x6DFB;&#x52A0;&#x4E00;&#x4E2A;&#x4E2D;&#x95F4;&#x8282;&#x70B9;&#x505A;&#x6839;&#xFF0C;&#x4E24;&#x9897;&#x6811;&#x4F5C;&#x4E3A;&#x5DE6;&#x53F3;&#x5B50;&#x6811;&#xFF0C;&#x6811;&#x6839;&#x503C;&#x662F;&#x4E24;&#x9897;&#x5B50;&#x6811;&#x7684;&#x6839;&#x503C;&#x7684;&#x548C;&#x3002;&#x5220;&#x6389;&#x4E24;&#x9897;&#x5B50;&#x6811;&#xFF0C;&#x628A;&#x65B0;&#x7684;&#x4E2D;&#x95F4;&#x8282;&#x70B9;&#x505A;&#x6839;&#x7684;&#x5B50;&#x6811;&#x6DFB;&#x52A0;&#x8FDB;&#x96C6;&#x5408;&#x3002;<br>
c. &#x91CD;&#x590D;&#x8FC7;&#x7A0B;b&#xFF0C;&#x76F4;&#x5230;&#x96C6;&#x5408;&#x91CC;&#x53EA;&#x6709;&#x4E00;&#x9897;&#x6811;&#x3002;</p>
</section><section vertical="true" lineno="86" class="slide " data-line="86" data-h="4" data-v="3">
<p>&#x7531;&#x6B64;&#x6211;&#x4EEC;&#x53EF;&#x77E5;&#xFF0C;&#x71B5;&#x4EE3;&#x8868;&#x4E86;&#x5BF9;&#x4FE1;&#x6E90;&#x8FDB;&#x884C;&#x6700;&#x4F18;&#x7F16;&#x7801;&#x65F6;&#x7684;&#x671F;&#x671B;&#x7F16;&#x7801;&#x957F;&#x5EA6;&#x3002;&#x53CD;&#x8FC7;&#x6765;&#x770B;&#xFF0C;&#x5982;&#x679C;&#x5C06;&#x8FD9;&#x4E2A;&#x4FE1;&#x6E90;&#x7528;&#x4E00;&#x4E2A;&#x968F;&#x673A;&#x53D8;&#x91CF;&#x6765;&#x8868;&#x793A;&#xFF0C;&#x82E5;&#x8BE5;&#x968F;&#x673A;&#x53D8;&#x91CF;&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x8D8A;&#x9AD8;&#xFF08;&#x4EA7;&#x751F;&#x7684;&#x4FE1;&#x606F;&#x79CD;&#x7C7B;&#x8D8A;&#x591A;&#x3001;&#x5404;&#x79CD;&#x7C7B;&#x51FA;&#x73B0;&#x7684;&#x6982;&#x7387;&#x8D8A;&#x5E73;&#x5747;&#xFF09;&#xFF0C;&#x5219;&#x9700;&#x8981;&#x7528;&#x6765;&#x7F16;&#x7801;&#x8BE5;&#x4FE1;&#x6E90;&#x7684;&#x671F;&#x671B;&#x7F16;&#x7801;&#x957F;&#x5EA6;&#x4E5F;&#x4F1A;&#x8D8A;&#x5927;&#xFF0C;&#x53CD;&#x4E4B;&#x5219;&#x8D8A;&#x77ED;&#x3002;&#x56E0;&#x800C;&#xFF0C;&#x71B5;&#x53C8;&#x53EF;&#x4EE5;&#x8868;&#x793A;&#x968F;&#x673A;&#x53D8;&#x91CF;&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x7A0B;&#x5EA6;&#x3002;</p>
</section><section vertical="true" lineno="88" class="slide " data-line="88" data-h="4" data-v="4">
<p>&#x4F8B;&#x5982;&#xFF0C;&#x4E00;&#x4E2A;&#x53D6;&#x503C;&#x4E3A;0&#x6216;1&#x7684;&#x968F;&#x673A;&#x53D8;&#x91CF;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>&#xFF0C;&#x8BA1;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p=P(X=1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span>&#xFF0C;&#x6839;&#x636E;&#x71B5;&#x7684;&#x5B9A;&#x4E49;&#xFF0C;&#x6709;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mi>p</mi><mi>log</mi><mo>&#x2061;</mo><mi>p</mi><mo>&#x2212;</mo><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>p</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>&#x2212;</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(X)=-p\log{p}-(1-p)\log{(1-p)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">&#x2212;</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">p</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></span></span><br>
&#x968F;&#x7740;p&#x7684;&#x53D8;&#x5316;&#xFF0C;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span>&#x7684;&#x53D8;&#x5316;&#x66F2;&#x7EBF;&#x5982;&#x4E0B;&#x56FE;&#xFF1A;<br>
<img src="https://img-blog.csdnimg.cn/img_convert/4c08857bcb5408dddcd956202a34da95.png#pic_center" alt="&#x4E8C;&#x503C;&#x968F;&#x673A;&#x53D8;&#x91CF;&#x7684;&#x71B5;.png"><br>
&#x4ECE;&#x4E0A;&#x56FE;&#x53EF;&#x77E5;&#xFF0C;&#x5F53;p=0&#x6216;&#x8005;p=1&#x65F6;&#xFF0C;&#x968F;&#x673A;&#x53D8;&#x91CF;X&#x7684;&#x53D6;&#x503C;&#x662F;&#x786E;&#x5B9A;&#x7684;&#xFF0C;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x6700;&#x5C0F;&#xFF0C;p=0.5&#x65F6;&#xFF0C;X&#x53D6;&#x503C;&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x6700;&#x5927;&#xFF0C;&#x6B64;&#x65F6;H(X)=1.</p>
</section><section vertical="true" lineno="96" class="slide " data-line="96" data-h="4" data-v="5">
<p>&#x540C;&#x6837;&#xFF0C;&#x6211;&#x4EEC;&#x53EF;&#x4EE5;&#x7528;&#x968F;&#x673A;&#x53D8;&#x91CF;X&#x3001;Y&#x3001;Z&#x5206;&#x522B;&#x8868;&#x793A;&#x63B7;&#x786C;&#x5E01;&#x3001;&#x63B7;&#x9AB0;&#x5B50;&#x548C;&#x4ECE;54&#x5F20;&#x6251;&#x514B;&#x724C;&#x4E2D;&#x968F;&#x673A;&#x62BD;&#x53D6;&#x4E00;&#x5F20;&#x7684;&#x7ED3;&#x679C;&#x3002;&#x663E;&#x7136;X&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x6700;&#x5C0F;&#xFF0C;Z&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x6700;&#x5927;&#x3002;&#x5B83;&#x4EEC;&#x7684;&#x71B5;&#x5219;&#x5B58;&#x5728;&#x5982;&#x4E0B;&#x7684;&#x5173;&#x7CFB;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>log</mi><mo>&#x2061;</mo><mn>6</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>log</mi><mo>&#x2061;</mo><mn>54</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
&amp; H(X)&lt;H(Y)&lt;H(Z) \\
&amp; H(X)=1 \\
&amp; H(Y)=\log{6} \\
&amp; H(Z)=\log{54}
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6em;vertical-align:-2.7500000000000004em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.25em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span><span style="top:-3.75em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span><span style="top:-2.249999999999999em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span><span style="top:-0.7499999999999996em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:2.7500000000000004em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em;"><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span></span></span><span style="top:-2.4099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span style="top:-0.9099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">54</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:2.7500000000000004em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
</section><section vertical="true" lineno="106" class="slide " data-line="106" data-h="4" data-v="6">
<p>&#x6211;&#x4EEC;&#x7528;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">&#x2223;</mi><mi>X</mi><mi mathvariant="normal">&#x2223;</mi></mrow><annotation encoding="application/x-tex">|X|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span></span></span></span>&#x6765;&#x8BA1;&#x53D8;&#x91CF;X&#x7684;&#x53D6;&#x503C;&#x4E2A;&#x6570;&#xFF0C;&#x53C8;&#x79F0;&#x4E3A;&#x53D8;&#x91CF;&#x7684;&#x52BF;&#x3002;&#x6211;&#x4EEC;&#x6709;&#x4EE5;&#x4E0B;&#x71B5;&#x7684;&#x57FA;&#x672C;&#x6027;&#x8D28;&#xFF0C;&#x5176;&#x4E2D;&#x6027;&#x8D28;(2)&#x5E38;&#x88AB;&#x79F0;&#x4E3A;<strong>&#x6700;&#x5927;&#x71B5;&#x539F;&#x7406;</strong>&#xFF1A;</p>
<blockquote>
<p>&#x71B5;&#x7684;&#x57FA;&#x672C;&#x6027;&#x8D28;&#xFF1A;<br>
(1) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x2265;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">H(X)\geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>;<br>
(2) <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x2264;</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi mathvariant="normal">&#x2223;</mi><mi>X</mi><mi mathvariant="normal">&#x2223;</mi></mrow></mrow><annotation encoding="application/x-tex">H(X)\leq \log{|X|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2264;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span></span></span></span></span>&#xFF0C;&#x7B49;&#x53F7;&#x6210;&#x7ACB;&#x7684;&#x6761;&#x4EF6;&#x662F;&#x5BF9;X&#x7684;&#x6240;&#x6709;&#x53D6;&#x503C;x&#x90FD;&#x6709;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">&#x2223;</mi><mi>X</mi><mi mathvariant="normal">&#x2223;</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">P(X=x)=\frac{1}{|X|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.365108em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">&#x2223;</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mord mtight">&#x2223;</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p>
</blockquote>
<p><strong>&#x8BC1;&#x660E;&#xFF1A;</strong><br>
(1)&#x6839;&#x636E;&#x71B5;&#x7684;&#x5B9A;&#x4E49;&#xFF0C;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x2265;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">H(X)\geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>&#x663E;&#x7136;&#x6210;&#x7ACB;&#x3002;<br>
(2)log&#x4E3A;&#x4E0A;&#x51F9;&#x51FD;&#x6570;&#xFF0C;&#x6839;&#x636E;Jensen&#x4E0D;&#x7B49;&#x5F0F;&#x6709;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>&#x2264;</mo><mi>log</mi><mo>&#x2061;</mo><mrow><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi mathvariant="normal">&#x2223;</mi><mi>X</mi><mi mathvariant="normal">&#x2223;</mi></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
H(X) &amp;=\sum_X P(X)\log{\frac{1}{P(X)}} \\
&amp;\leq \log{\sum_X P(X)\frac{1}{P(X)}} \\
&amp;= \log{|X|}
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.331552em;vertical-align:-3.415776em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.915776em;"><span style="top:-5.915776em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"></span></span><span style="top:-0.565664em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:3.415776em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.915776em;"><span style="top:-5.915776em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2264;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span style="top:-0.565664em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:3.415776em;"><span></span></span></span></span></span></span></span></span></span></span></span><br>
&#x547D;&#x9898;&#x5F97;&#x8BC1;&#x3002;</p>
</section></section><section><section lineno="124" class="slide " data-line="124" data-h="5" data-v="0">
<h3 class="mume-header" id="113-%E8%81%94%E5%90%88%E7%86%B5-%E6%9D%A1%E4%BB%B6%E7%86%B5-%E4%BA%92%E4%BF%A1%E6%81%AF">1.1.3 &#x8054;&#x5408;&#x71B5;&#x3001;&#x6761;&#x4EF6;&#x71B5;&#x3001;&#x4E92;&#x4FE1;&#x606F;</h3>

<p>&#x8054;&#x5408;&#x71B5;&#x662F;&#x57FA;&#x4E8E;&#x8054;&#x5408;&#x6982;&#x7387;&#x5206;&#x5E03;&#x5BF9;&#x71B5;&#x7684;&#x63A8;&#x5E7F;&#x3002;&#x4E24;&#x4E2A;&#x79BB;&#x6563;&#x968F;&#x673A;&#x53D8;&#x91CF;X&#x548C;Y&#x7684;&#x8054;&#x5408;&#x71B5;&#x5B9A;&#x4E49;&#x4E3A;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>E</mi><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></msub><mrow><mo fence="true">[</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(2)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
H(X,Y) &amp;=E_{P(X,Y)}\left[\log{\frac{1}{P(X,Y)}}\right] \\
&amp;= -\sum_{X,Y} P(X,Y)\log{P(X,Y)} 
\end{aligned}
\tag{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.480479em;vertical-align:-2.4902395000000004em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9902394999999995em;"><span style="top:-4.9902394999999995em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-2.6902045em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:2.4902395000000004em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9902394999999995em;"><span style="top:-4.9902394999999995em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span><span style="top:-2.6902045em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:2.4902395000000004em;"><span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:5.480479em;vertical-align:-2.4902395000000004em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">2</span></span><span class="mord">)</span></span></span></span></span></span></p>
</section><section vertical="true" lineno="134" class="slide " data-line="134" data-h="5" data-v="1">
<p>&#x6761;&#x4EF6;&#x71B5;&#x662F;&#x57FA;&#x4E8E;&#x6761;&#x4EF6;&#x6982;&#x7387;&#x5206;&#x5E03;&#x5BF9;&#x71B5;&#x7684;&#x63A8;&#x5E7F;&#x3002;&#x968F;&#x673A;&#x53D8;&#x91CF;X&#x7684;&#x71B5;&#x662F;&#x7528;&#x5B83;&#x7684;&#x6982;&#x7387;&#x5206;&#x5E03;P(X)&#x6765;&#x5B9A;&#x4E49;&#x7684;&#x3002;&#x5982;&#x679C;&#x77E5;&#x9053;&#x53E6;&#x4E00;&#x4E2A;&#x968F;&#x673A;&#x53D8;&#x91CF;Y&#x7684;&#x53D6;&#x503C;&#x4E3A;y&#xFF0C;&#x90A3;&#x4E48;X&#x7684;&#x6761;&#x4EF6;&#x5206;&#x5E03;&#x5373;&#x4E3A;P(X|Y=y)&#x3002;&#x5229;&#x7528;&#x6B64;&#x6761;&#x4EF6;&#x5206;&#x5E03;&#x53EF;&#x5B9A;&#x4E49;&#x7ED9;&#x5B9A;Y=y&#x65F6;X&#x7684;&#x6761;&#x4EF6;&#x71B5;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>E</mi><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msub><mrow><mo fence="true">[</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(3)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
H(X|Y=y) &amp;=E_{P(X|Y=y)}\left[\log{\frac{1}{P(X|Y=y)}}\right] \\
&amp;= -\sum_X P(X|Y=y)\log{P(X|Y=y)}
\end{aligned}
\tag{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.344371000000001em;vertical-align:-2.4221855000000003em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9221855000000003em;"><span style="top:-4.922185500000001em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-2.6221505em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:2.4221855000000003em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9221855000000003em;"><span style="top:-4.922185500000001em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mord mtight">&#x2223;</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span><span class="mrel mtight">=</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span><span style="top:-2.6221505em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:2.4221855000000003em;"><span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:5.344371000000001em;vertical-align:-2.4221855000000003em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3</span></span><span class="mord">)</span></span></span></span></span></span></p>
</section><section vertical="true" lineno="143" class="slide " data-line="143" data-h="5" data-v="2">
<p>&#x71B5;H(X)&#x5EA6;&#x91CF;&#x7684;&#x662F;&#x968F;&#x673A;&#x53D8;&#x91CF;X&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#xFF0C;&#x6761;&#x4EF6;&#x71B5;H(X|Y=y)&#x5EA6;&#x91CF;&#x7684;&#x5219;&#x662F;&#x5DF2;&#x77E5;Y=y&#x540E;&#xFF0C;X&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x3002;<br>
&#x4E0A;&#x5F0F;(3)&#x4E2D;&#xFF0C;&#x5F53;y&#x53D8;&#x5316;&#x65F6;&#xFF0C;H(X|Y=y)&#x4E5F;&#x4F1A;&#x53D1;&#x751F;&#x6539;&#x53D8;&#xFF0C;&#x5F53;&#x77E5;&#x9053;Y&#x7684;&#x6982;&#x7387;&#x5206;&#x5E03;&#x540E;&#xFF0C;&#x53EF;&#x4EE5;&#x8BA1;&#x7B97;X&#x5173;&#x4E8E;Y&#x7684;&#x6761;&#x4EF6;&#x71B5;&#x7684;&#x671F;&#x671B;&#x503C;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>&#x2211;</mo><mrow><mi>y</mi><mo>&#x2208;</mo><msub><mi mathvariant="normal">&#x3A9;</mi><mi>Y</mi></msub></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mrow><mi>y</mi><mo>&#x2208;</mo><msub><mi mathvariant="normal">&#x3A9;</mi><mi>Y</mi></msub></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mi>Y</mi></munder><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable></mtd><mtd width="50%"></mtd><mtd><mtext>(4)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
H(X|Y) &amp;=\sum_{y\in \Omega_Y} P(Y=y)H(X|Y=y) \\
&amp;= -\sum_{y\in \Omega_Y} P(Y=y) \sum_X P(X|Y=y)\log{P(X|Y=y)} \\
&amp;= -\sum_Y \sum_X P(Y)P(X|Y)\log{P(X|Y)} \\
&amp;= -\sum_{X,Y} P(X,Y)\log{P(X|Y)}
\end{aligned}
\tag{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.985688000000001em;vertical-align:-5.242844000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.742844000000001em;"><span style="top:-7.742844000000002em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-4.962395000000001em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"></span></span><span style="top:-2.181945999999999em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"></span></span><span style="top:0.4623950000000001em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:5.242844000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.742844000000001em;"><span style="top:-7.742844000000002em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mrel mtight">&#x2208;</span><span class="mord mtight"><span class="mord mtight">&#x3A9;</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-4.962395000000001em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mrel mtight">&#x2208;</span><span class="mord mtight"><span class="mord mtight">&#x3A9;</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span style="top:-2.181945999999999em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span style="top:0.4623950000000001em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:5.242844000000001em;"><span></span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:10.985688000000001em;vertical-align:-5.242844000000001em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">4</span></span><span class="mord">)</span></span></span></span></span></span></p>
</section><section vertical="true" lineno="155" class="slide " data-line="155" data-h="5" data-v="3">
<p>H(X|Y)&#x79F0;&#x4E3A;&#x7ED9;&#x5B9A;Y&#x65F6;X&#x7684;&#x6761;&#x4EF6;&#x71B5;&#x3002;<br>
&#x6CE8;&#x610F;&#xFF1A;H(X|Y)&#x548C;H(X|Y=y)&#x4E0D;&#x4E00;&#x6837;&#xFF0C;&#x540E;&#x8005;&#x662F;&#x5DF2;&#x77E5;Y&#x53D6;&#x67D0;&#x4E00;&#x7279;&#x5B9A;&#x503C;&#x65F6;X&#x7684;&#x6761;&#x4EF6;&#x71B5;&#xFF0C;&#x5373;&#x5DF2;&#x77E5;Y=y&#x540E;&#xFF0C;X&#x5269;&#x4F59;&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x3002;&#x800C;&#x524D;&#x8005;&#x662F;&#x5728;&#x672A;&#x77E5;Y&#x7684;&#x53D6;&#x503C;&#x65F6;&#xFF0C;&#x5BF9;&#x89C2;&#x6D4B;&#x5230;Y&#x7684;&#x53D6;&#x503C;&#x540E;X&#x5269;&#x4F59;&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x7684;&#x671F;&#x671B;&#x503C;&#x3002;&#x5C24;&#x5176;&#x503C;&#x5F97;&#x6CE8;&#x610F;&#x7684;&#x662F;&#xFF0C;H(X|Y=y)&#x53EF;&#x80FD;&#x6BD4;H(X)&#x5927;&#xFF0C;&#x5373;&#x77E5;&#x9053;Y&#x7684;&#x67D0;&#x4E2A;&#x5177;&#x4F53;&#x53D6;&#x503C;&#x540E;&#xFF0C;&#x6709;&#x53EF;&#x80FD;&#x589E;&#x5927;&#x5BF9;X&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x3002;&#x800C;H(X|Y)&#x6C38;&#x8FDC;&#x4E0D;&#x4F1A;&#x6BD4;H(X)&#x5927;&#xFF0C;&#x5373;&#x5E73;&#x5747;&#x6765;&#x8BF4;&#xFF0C;&#x77E5;&#x9053;Y&#x4E0D;&#x4F1A;&#x589E;&#x52A0;X&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x3002;&#x4E0B;&#x9762;&#x7ED9;&#x51FA;&#x4E00;&#x4E2A;&#x5177;&#x4F53;&#x7684;&#x4F8B;&#x5B50;&#x52A0;&#x4EE5;&#x6BD4;&#x8F83;&#xFF1A;</p>
</section><section vertical="true" lineno="158" class="slide " data-line="158" data-h="5" data-v="4">
<p>&#x8BBE;&#x5DF2;&#x77E5;&#x8054;&#x5408;&#x5206;&#x5E03;P(X,Y)&#x53CA;&#x5176;&#x8FB9;&#x7F18;&#x5206;&#x5E03;P(X)&#x548C;P(Y)&#x5982;&#x4E0B;&#x8868;&#x6240;&#x793A;&#xFF1A;</p>
<table>
<thead>
<tr>
<th style="text-align:center"></th>
<th style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></th>
<th style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></th>
<th style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">y_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:center">0</td>
<td style="text-align:center">3/4</td>
<td style="text-align:center">3/4</td>
</tr>
<tr>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">y_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:center">1/8</td>
<td style="text-align:center">1/8</td>
<td style="text-align:center">1/4</td>
</tr>
<tr>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></td>
<td style="text-align:center">1/8</td>
<td style="text-align:center">7/8</td>
<td style="text-align:center"></td>
</tr>
</tbody>
</table>
<p>&#x4ECE;&#x800C;&#x53EF;&#x5F97;&#x51FA;&#xFF1A;</p>
</section><section vertical="true" lineno="166" class="slide " data-line="166" data-h="5" data-v="5">
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>&#x2212;</mo><mfrac><mn>7</mn><mn>8</mn></mfrac><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>7</mn><mn>8</mn></mfrac><mo>=</mo><mn>0.544</mn><mo separator="true">;</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mn>0</mn><mi>log</mi><mo>&#x2061;</mo><mn>0</mn><mo>&#x2212;</mo><mn>1</mn><mi>log</mi><mo>&#x2061;</mo><mn>1</mn><mo>=</mo><mn>0</mn><mo separator="true">;</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>&#x2212;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>&#x2212;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>1</mn><mo separator="true">;</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.25</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
&amp; H(X)= -\frac{1}{8}\log{\frac{1}{8}}-\frac{7}{8}\log{\frac{7}{8}}=0.544; \\
&amp; H(X|Y=y_1)= -0\log{0}-1\log{1}=0; \\
&amp; H(X|Y=y_2)= -\frac{1}{2}\log{\frac{1}{2}}-\frac{1}{2}\log{\frac{1}{2}}=1; \\
&amp; H(X|Y)= \frac{3}{4}H(X|Y=y_1)+\frac{1}{4}H(X|Y=y_2)=0.25
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.422320000000001em;vertical-align:-3.96116em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.4611600000000005em;"><span style="top:-6.4611600000000005em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"></span></span><span style="top:-4.63516em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"></span></span><span style="top:-2.6537199999999994em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"></span></span><span style="top:-0.34628000000000003em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:3.96116em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.4611600000000005em;"><span style="top:-6.4611600000000005em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0.544</span><span class="mpunct">;</span></span></span><span style="top:-4.63516em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mord">0</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">0</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord">1</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span><span class="mpunct">;</span></span></span><span style="top:-2.6537199999999994em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span><span class="mpunct">;</span></span></span><span style="top:-0.34628000000000003em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0.25</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:3.96116em;"><span></span></span></span></span></span></span></span></span></span></span></span><br>
&#x53EF;&#x4EE5;&#x770B;&#x5230;&#xFF1A;&#x89C2;&#x6D4B;&#x5230;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">Y=y_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x540E;&#xFF0C;&#x53EF;&#x4F7F;X&#x7684;&#x71B5;&#x51CF;&#x5C0F;&#xFF1B;&#x89C2;&#x6D4B;&#x5230;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Y=y_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x540E;&#xFF0C;&#x53CD;&#x800C;&#x4F7F;X&#x7684;&#x71B5;&#x589E;&#x5927;&#xFF1B;&#x4F46;&#x5E73;&#x5747;&#x800C;&#x8A00;&#xFF0C;&#x5BF9;Y&#x7684;&#x89C2;&#x6D4B;&#x4F7F;X&#x7684;&#x71B5;&#x51CF;&#x5C0F;&#x3002;</p>
</section><section vertical="true" lineno="176" class="slide " data-line="176" data-h="5" data-v="6">
<p>&#x7531;&#x6B64;&#xFF0C;&#x6211;&#x4EEC;&#x5B9A;&#x4E49;&#x4E92;&#x4FE1;&#x606F;&#x91CF;&#x4E3A;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">;</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(X;Y)=H(X)-H(X|Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></span><br>
&#x79F0;&#x4E3A;Y&#x5173;&#x4E8E;X&#x7684;&#x4FE1;&#x606F;&#xFF0C;&#x8868;&#x793A;Y&#x4E2D;&#x5305;&#x542B;&#x591A;&#x5C11;&#x5173;&#x4E8E;X&#x7684;&#x4FE1;&#x606F;&#x3002;&#x5F88;&#x5BB9;&#x6613;&#x8BC1;&#x660E;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">;</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mi>Y</mi><mo separator="true">;</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(X;Y)=I(Y;X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span>&#xFF0C;&#x56E0;&#x6B64;&#x53C8;&#x79F0;&#x4E4B;&#x4E3A;X&#x548C;Y&#x4E4B;&#x95F4;&#x7684;&#x4E92;&#x4FE1;&#x606F;&#x3002;</p>
</section><section vertical="true" lineno="182" class="slide " data-line="182" data-h="5" data-v="7">
<p><strong>&#x4E92;&#x4FE1;&#x606F;&#x7684;&#x6027;&#x8D28;1&#xFF1A;</strong><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">;</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mi>Y</mi><mo separator="true">;</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(X;Y)=I(Y;X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></span></p>
<p><strong>&#x8BC1;&#x660E;&#xFF1A;</strong><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">;</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
I(X;Y) &amp;= H(X)-H(X|Y) \\
&amp;= -\sum_X P(X)\log{P(X)}+ \sum_{X,Y} P(X,Y)\log{P(X|Y)} \\
&amp;= -\sum_{X,Y} P(X,Y)\log{P(X)}+ \sum_{X,Y} P(X,Y)\log{P(X|Y)} \\
&amp;= \sum_{X,Y} P(X,Y)\log{\frac{P(X|Y)}{P(X)}} \\
&amp;= \sum_{X,Y} P(X,Y)\log{\frac{P(X,Y)}{P(X)P(Y)}}
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:13.375786000000002em;vertical-align:-6.437893000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.937893000000001em;"><span style="top:-9.524893em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-7.814888em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-5.034438999999999em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-1.8769949999999993em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:1.2804490000000006em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:6.437893000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.937893000000001em;"><span style="top:-9.524893em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-7.814888em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span style="top:-5.034438999999999em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span style="top:-1.8769949999999993em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span style="top:1.2804490000000006em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:6.437893000000001em;"><span></span></span></span></span></span></span></span></span></span></span></span><br>
&#x540C;&#x7406;&#x53EF;&#x5F97;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>Y</mi><mo separator="true">;</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">I(Y;X)=\sum_{X,Y} P(X,Y)\log{\frac{P(X,Y)}{P(X)P(Y)}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.857444em;vertical-align:-1.430444em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span><br>
&#x56E0;&#x6B64;&#xFF0C;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">;</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mi>Y</mi><mo separator="true">;</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(X;Y)=I(Y;X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span>&#x5F97;&#x8BC1;&#x3002;</p>
</section><section vertical="true" lineno="203" class="slide " data-line="203" data-h="5" data-v="8">
<p><strong>&#x71B5;&#x7684;&#x94FE;&#x89C4;&#x5219;&#xFF1A;</strong><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi><mo stretchy="false">(</mo><mi>Y</mi><mi mathvariant="normal">&#x2223;</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(5)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">H(X,Y)=H(X)+H(Y|X)=H(Y)+H(X|Y) \tag{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p><strong>&#x8BC1;&#x660E;&#xFF1A;</strong><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mi mathvariant="normal">&#x2223;</mi><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mi mathvariant="normal">&#x2223;</mi><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi><mo stretchy="false">(</mo><mi>Y</mi><mi mathvariant="normal">&#x2223;</mi><mi>X</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
H(X,Y) &amp;=-\sum_{X,Y} P(X,Y)\log{P(X,Y)} \\
&amp;= -\sum_{X,Y} P(X,Y)\log{P(X)} -\sum_{X,Y} P(X,Y)\log{P(Y|X)} \\
&amp;= -\sum_{X} P(X)\log{P(X)}-\sum_{X,Y} P(X,Y)\log{P(Y|X)} \\
&amp;= H(X)+H(Y|X)
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.841347000000003em;vertical-align:-4.670673500000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.1706735em;"><span style="top:-7.170673500000001em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-4.390224500000001em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"></span></span><span style="top:-1.6097755000000002em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"></span></span><span style="top:0.9606685000000006em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:4.670673500000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.1706735em;"><span style="top:-7.170673500000001em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span style="top:-4.390224500000001em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span><span style="top:-1.6097755000000002em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span><span style="top:0.9606685000000006em;"><span class="pstrut" style="height:3.0500050000000005em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:4.670673500000001em;"><span></span></span></span></span></span></span></span></span></span></span></span><br>
&#x540C;&#x7406;&#x53EF;&#x8BC1;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X,Y)=H(Y)+H(X|Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></p>
</section><section vertical="true" lineno="219" class="slide " data-line="219" data-h="5" data-v="9">
<p><strong>&#x4E92;&#x4FE1;&#x606F;&#x7684;&#x6027;&#x8D28;2&#xFF1A;</strong><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(6)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">I(X,Y)=H(X)+H(Y)-H(X,Y) \tag{6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span><span class="tag"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">6</span></span><span class="mord">)</span></span></span></span></span></span></p>
<p><strong>&#x8BC1;&#x660E;&#xFF1A;</strong><br>
&#x7B49;&#x5F0F;&#x5DE6;&#x8FB9;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(X,Y)=H(X)-H(X|Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></span><br>
&#x7B49;&#x5F0F;&#x53F3;&#x8FB9;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mtext>&#x71B5;&#x7684;&#x94FE;&#x89C4;&#x5219;</mtext><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
H(X)+H(Y)-H(X,Y) &amp;=H(X)+H(Y)-(H(Y)+H(X|Y)) (&#x71B5;&#x7684;&#x94FE;&#x89C4;&#x5219;) \\
&amp;=H(X)-H(X|Y)
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000000000000004em;vertical-align:-1.2500000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">))</span><span class="mopen">(</span><span class="mord cjk_fallback">&#x71B5;&#x7684;&#x94FE;&#x89C4;&#x5219;</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span><br>
&#x4ECE;&#x800C;&#x7B49;&#x5F0F;&#x6210;&#x7ACB;&#x3002;</p>
</section><section vertical="true" lineno="238" class="slide " data-line="238" data-h="5" data-v="10">
<p>&#x8054;&#x5408;&#x71B5;&#x3001;&#x6761;&#x4EF6;&#x71B5;&#x548C;&#x4E92;&#x4FE1;&#x606F;&#x4E4B;&#x95F4;&#x7684;&#x5173;&#x7CFB;&#xFF0C;&#x53EF;&#x7528;&#x5982;&#x4E0B;&#x6587;&#x6C0F;&#x56FE;&#x6765;&#x8868;&#x793A;&#x5B83;&#x4EEC;&#x4E4B;&#x95F4;&#x7684;&#x5173;&#x7CFB;&#xFF1A;<br>
<img src="https://img-blog.csdnimg.cn/img_convert/ec9938373efba429373b8994fb03dfb3.png#pic_center" alt="&#x8054;&#x5408;&#x71B5;&#x3001;&#x6761;&#x4EF6;&#x71B5;&#x548C;&#x4E92;&#x4FE1;&#x606F;&#x4E4B;&#x95F4;&#x7684;&#x5173;&#x7CFB;.png"></p>
</section></section><section><section lineno="242" class="slide " data-line="242" data-h="6" data-v="0">
<h3 class="mume-header" id="114-%E4%BA%A4%E5%8F%89%E7%86%B5%E5%92%8C%E7%9B%B8%E5%AF%B9%E7%86%B5kl%E6%95%A3%E5%BA%A6">1.1.4 &#x4EA4;&#x53C9;&#x71B5;&#x548C;&#x76F8;&#x5BF9;&#x71B5;&#xFF08;KL&#x6563;&#x5EA6;&#xFF09;</h3>

<p>&#x5728;1.1.2&#x8282;&#x4ECB;&#x7ECD;&#x71B5;&#x7684;&#x6982;&#x5FF5;&#x65F6;&#xFF0C;&#x4ECB;&#x7ECD;&#x4E86;&#x71B5;&#x7684;&#x671F;&#x671B;&#x7F16;&#x7801;&#x957F;&#x5EA6;&#x7684;&#x610F;&#x4E49;&#x3002;&#x4EA4;&#x53C9;&#x71B5;&#x7684;&#x6982;&#x5FF5;&#x4E5F;&#x53EF;&#x4EE5;&#x4ECE;&#x671F;&#x671B;&#x7F16;&#x7801;&#x957F;&#x5EA6;&#x7684;&#x610F;&#x4E49;&#x51FA;&#x53D1;&#x8FDB;&#x884C;&#x7406;&#x89E3;&#x3002;<br>
&#x82E5;&#x4FE1;&#x6E90;X&#x7684;&#x7406;&#x8BBA;&#x6982;&#x7387;&#x5206;&#x5E03;&#x4E3A;Q(X)&#xFF0C;&#x4F46;&#x5176;&#x5B9E;&#x9645;&#x6982;&#x7387;&#x5206;&#x5E03;&#x4E3A;P(X)&#xFF0C;&#x5219;&#x4F7F;&#x7528;&#x7406;&#x8BBA;&#x6982;&#x7387;&#x5206;&#x5E03;&#x6784;&#x5EFA;&#x7684;&#x970D;&#x592B;&#x66FC;&#x7F16;&#x7801;&#x5728;&#x5B9E;&#x9645;&#x6982;&#x7387;&#x5206;&#x5E03;&#x4E0B;&#x7684;&#x671F;&#x671B;&#x7F16;&#x7801;&#x957F;&#x5EA6;&#x5373;&#x4E3A;&#x4EA4;&#x53C9;&#x71B5;&#xFF0C;&#x5B9A;&#x4E49;&#x4E3A;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>P</mi><mo separator="true">,</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mrow><mo fence="true">[</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>1</mn><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac><mo fence="true">]</mo></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(P,Q)=E_{P(X)} \left[ \log{\frac{1}{Q(X)}}\right] =-\sum_X P(X)\log{Q(X)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">Q</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">P</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.344341em;vertical-align:-1.294336em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></span></span></p>
</section><section vertical="true" lineno="249" class="slide " data-line="249" data-h="6" data-v="1">
<p>&#x76F8;&#x5BF9;&#x71B5;&#x5219;&#x5B9A;&#x4E49;&#x4E3A;&#x4EA4;&#x53C9;&#x71B5;&#x4E0E;&#x71B5;&#x4E4B;&#x5DEE;&#xFF0C;&#x5373;&#x6309;&#x7167;&#x4FE1;&#x6E90;&#x7684;&#x7406;&#x8BBA;&#x6982;&#x7387;&#x5206;&#x5E03;Q&#x8BBE;&#x8BA1;&#x7684;&#x6700;&#x4F18;&#x7F16;&#x7801;&#x7684;&#x671F;&#x671B;&#x7801;&#x957F;&#x4F1A;&#x6BD4;&#x6309;&#x7167;&#x5B9E;&#x9645;&#x6982;&#x7387;&#x5206;&#x5E03;P&#x8BBE;&#x8BA1;&#x7684;&#x6700;&#x4F18;&#x7F16;&#x7801;&#x7684;&#x671F;&#x671B;&#x7801;&#x957F;&#x591A;&#x51E0;&#x4E2A;&#x6BD4;&#x7279;&#x3002;&#x5176;&#x5B9A;&#x4E49;&#x5982;&#x4E0B;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>K</mi><mi>L</mi><mo stretchy="false">(</mo><mi>P</mi><mo separator="true">,</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">KL(P,Q)=\sum_X P(X)\log{\frac{P(X)}{Q(X)}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">Q</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.721336em;vertical-align:-1.294336em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span><br>
&#x5176;&#x4E2D;&#x7EA6;&#x5B9A;&#xFF1A;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mi>log</mi><mo>&#x2061;</mo><mfrac><mn>0</mn><mi>q</mi></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0\log{\frac{0}{q}}=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.326216em;vertical-align:-0.481108em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.481108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>&#xFF1B;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">&#x2200;</mi><mi>p</mi><mo>&gt;</mo><mn>0</mn><mo>:</mo><mi>p</mi><mi>log</mi><mo>&#x2061;</mo><mfrac><mi>p</mi><mn>0</mn></mfrac><mo>=</mo><mi mathvariant="normal">&#x221E;</mi></mrow><annotation encoding="application/x-tex">\forall p&gt;0: p\log{\frac{p}{0}}=\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">&#x2200;</span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0925em;vertical-align:-0.345em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7475em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.446108em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">p</span></span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord">&#x221E;</span></span></span></span>.<br>
KL(P,Q)&#x53C8;&#x79F0;&#x4E3A;P(X)&#x548C;Q(X)&#x4E4B;&#x95F4;&#x7684;Kullback-Leibler&#x8DDD;&#x79BB;&#xFF0C;&#x6216;KL&#x6563;&#x5EA6;&#x3002;&#x4F46;&#x4E25;&#x683C;&#x6765;&#x8BB2;&#xFF0C;&#x5B83;&#x5E76;&#x4E0D;&#x662F;&#x4E00;&#x4E2A;&#x771F;&#x6B63;&#x610F;&#x4E49;&#x7684;&#x8DDD;&#x79BB;&#xFF0C;&#x56E0;&#x4E3A;&#x5176;&#x4E0D;&#x6EE1;&#x8DB3;&#x5BF9;&#x79F0;&#x6027;&#x3002;</p>
</section><section vertical="true" lineno="256" class="slide " data-line="256" data-h="6" data-v="2">
<blockquote>
<p><strong>&#x4FE1;&#x606F;&#x4E0D;&#x7B49;&#x5F0F;&#xFF1A;</strong><br>
&#x8BBE;P(X)&#x548C;Q(X)&#x4E3A;&#x5B9A;&#x4E49;&#x5728;&#x968F;&#x673A;&#x53D8;&#x91CF;X&#x7684;&#x72B6;&#x6001;&#x7A7A;&#x95F4;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="normal">&#x3A9;</mi><mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Omega_X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">&#x3A9;</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x4E0A;&#x7684;&#x4E24;&#x4E2A;&#x6982;&#x7387;&#x5206;&#x5E03;&#xFF0C;&#x5219;&#x6709;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>K</mi><mi>L</mi><mo stretchy="false">(</mo><mi>P</mi><mo separator="true">,</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>&#x2265;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">KL(P,Q)\geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">Q</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span><br>
&#x5176;&#x4E2D;&#xFF0C;&#x5F53;&#x4E14;&#x4EC5;&#x5F53;P=Q&#xFF0C;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">&#x2200;</mi><mi>x</mi><mo>&#x2208;</mo><msub><mi mathvariant="normal">&#x3A9;</mi><mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\forall x \in \Omega_X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord">&#x2200;</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2208;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord">&#x3A9;</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>&#x65F6;&#x7B49;&#x53F7;&#x6210;&#x7ACB;&#x3002;</p>
</blockquote>
</section><section vertical="true" lineno="263" class="slide " data-line="263" data-h="6" data-v="3">
<p><strong>&#x8BC1;&#x660E;&#xFF1A;</strong><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>K</mi><mi>L</mi><mo stretchy="false">(</mo><mi>P</mi><mo separator="true">,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>&#x2265;</mo><mo>&#x2212;</mo><mi>log</mi><mo>&#x2061;</mo><mrow><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mfrac><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo stretchy="false">(</mo><mtext>&#x6839;&#x636E;&#x4E0B;&#x51F8;&#x51FD;&#x6570;&#x7684;</mtext><mi>J</mi><mi>e</mi><mi>n</mi><mi>s</mi><mi>e</mi><mi>n</mi><mtext>&#x4E0D;&#x7B49;&#x5F0F;</mtext><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>&#x2212;</mo><mi>log</mi><mo>&#x2061;</mo><mrow><munder><mo>&#x2211;</mo><mi>X</mi></munder><mi>Q</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
KL(P,Q) &amp;= \sum_X P(X)\log{\frac{P(X)}{Q(X)}} \\
&amp;= -\sum_X P(X)\log{\frac{Q(X)}{P(X)}} \\
&amp;\geq -\log{\sum_X P(X)\frac{Q(X)}{P(X)}} (&#x6839;&#x636E;&#x4E0B;&#x51F8;&#x51FD;&#x6570;&#x7684;Jensen&#x4E0D;&#x7B49;&#x5F0F;) \\
&amp;= -\log{\sum_X Q(X)} \\
&amp;= 0
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:13.208349em;vertical-align:-6.354174500000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.854174499999999em;"><span style="top:-8.8541745em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">Q</span><span class="mclose">)</span></span></span><span style="top:-5.832838500000001em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-2.8115025000000005em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:-0.16716149999999996em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span><span style="top:2.2671745000000008em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:6.354174500000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.854174499999999em;"><span style="top:-8.8541745em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span style="top:-5.832838500000001em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span style="top:-2.8115025000000005em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mopen">(</span><span class="mord cjk_fallback">&#x6839;&#x636E;&#x4E0B;&#x51F8;&#x51FD;&#x6570;&#x7684;</span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mord mathnormal">se</span><span class="mord mathnormal">n</span><span class="mord cjk_fallback">&#x4E0D;&#x7B49;&#x5F0F;</span><span class="mclose">)</span></span></span><span style="top:-0.16716149999999996em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">&#x2212;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0500050000000003em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.294336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">Q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span><span style="top:2.2671745000000008em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:6.354174500000001em;"><span></span></span></span></span></span></span></span></span></span></span></span><br>
&#x4FE1;&#x606F;&#x4E0D;&#x7B49;&#x5F0F;&#x5F97;&#x8BC1;&#x3002;<br>
&#x5229;&#x7528;KL&#x6563;&#x5EA6;&#x53EF;&#x4EE5;&#x5EA6;&#x91CF;&#x4E24;&#x4E2A;&#x6982;&#x7387;&#x5206;&#x5E03;&#x4E4B;&#x95F4;&#x7684;&#x5DEE;&#x5F02;&#x3002;</p>
</section></section><section><section lineno="277" class="slide " data-line="277" data-h="7" data-v="0">
<h3 class="mume-header" id="115-%E4%BA%92%E4%BF%A1%E6%81%AF%E4%B8%8E%E5%8F%98%E9%87%8F%E7%8B%AC%E7%AB%8B">1.1.5 &#x4E92;&#x4FE1;&#x606F;&#x4E0E;&#x53D8;&#x91CF;&#x72EC;&#x7ACB;</h3>

<p>&#x4ECE;1.1.3&#x8282;&#x7ED9;&#x51FA;&#x7684;&#x8054;&#x5408;&#x71B5;&#x3001;&#x6761;&#x4EF6;&#x71B5;&#x4E0E;&#x4E92;&#x4FE1;&#x606F;&#x4E4B;&#x95F4;&#x5173;&#x7CFB;&#x7684;&#x6587;&#x6C0F;&#x56FE;&#x53EF;&#x4EE5;&#x770B;&#x51FA;&#xFF1A;&#x5BF9;&#x4E8E;&#x968F;&#x673A;&#x53D8;&#x91CF;X&#x548C;Y&#xFF0C;&#x5F53;&#x4E92;&#x4FE1;&#x606F;I(X,Y)=0&#x65F6;&#xFF0C;X&#x548C;Y&#x76F8;&#x4E92;&#x72EC;&#x7ACB;&#xFF1B;&#x4E14;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x2264;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X|Y)\leq H(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2264;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span>&#xFF0C;&#x7B49;&#x53F7;&#x4E5F;&#x5728;X&#x548C;Y&#x72EC;&#x7ACB;&#x65F6;&#x6210;&#x7ACB;&#x3002;&#x6211;&#x4EEC;&#x4E5F;&#x53EF;&#x4EE5;&#x7ED9;&#x51FA;&#x4E25;&#x683C;&#x8BC1;&#x660E;&#x3002;<br>
<strong>&#x8BC1;&#x660E;&#xFF1A;</strong><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>&#x2211;</mo><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></munder><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mi>log</mi><mo>&#x2061;</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">(</mo><mn>1.1.3</mn><mtext>&#x8282;&#x8BC1;&#x660E;&#x4E92;&#x4FE1;&#x606F;&#x6027;&#x8D28;</mtext><mn>1</mn><mtext>&#x7684;&#x4E2D;&#x95F4;&#x7ED3;&#x8BBA;</mtext><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>K</mi><mi>L</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>K</mi><mi>L</mi><mtext>&#x6563;&#x5EA6;&#x7684;&#x5B9A;&#x4E49;</mtext><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
I(X,Y) &amp;=\sum_{X,Y} P(X,Y)\log{\frac{P(X,Y)}{P(X)P(Y)}} (1.1.3&#x8282;&#x8BC1;&#x660E;&#x4E92;&#x4FE1;&#x606F;&#x6027;&#x8D28;1&#x7684;&#x4E2D;&#x95F4;&#x7ED3;&#x8BBA;) \\
&amp;=KL(P(X,Y),P(x)P(Y)) (KL&#x6563;&#x5EA6;&#x7684;&#x5B9A;&#x4E49;) \\
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.657444em;vertical-align:-2.078722em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.578722em;"><span style="top:-4.578722em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-2.0082780000000002em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:2.078722em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.578722em;"><span style="top:-4.578722em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">&#x2211;</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.430444em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mopen">(</span><span class="mord">1.1.3</span><span class="mord cjk_fallback">&#x8282;&#x8BC1;&#x660E;&#x4E92;&#x4FE1;&#x606F;&#x6027;&#x8D28;</span><span class="mord">1</span><span class="mord cjk_fallback">&#x7684;&#x4E2D;&#x95F4;&#x7ED3;&#x8BBA;</span><span class="mclose">)</span></span></span><span style="top:-2.0082780000000002em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">))</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord mathnormal">L</span><span class="mord cjk_fallback">&#x6563;&#x5EA6;&#x7684;&#x5B9A;&#x4E49;</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:2.078722em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
</section><section vertical="true" lineno="287" class="slide " data-line="287" data-h="7" data-v="1">
<p>&#x7531;KL&#x6563;&#x5EA6;&#x5927;&#x4E8E;&#x7B49;&#x4E8E;0&#x53EF;&#x5F97;&#xFF1A;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x2265;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">I(X,Y)\geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>&#xFF0C;&#x5F53;&#x4E14;&#x4EC5;&#x5F53;P(X,Y)=P(X)P(Y)&#x65F6;&#x7B49;&#x53F7;&#x6210;&#x7ACB;&#xFF0C;&#x5373;X&#x4E0E;Y&#x76F8;&#x4E92;&#x72EC;&#x7ACB;&#x3002;<br>
&#x7531;&#x4E8E;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x2265;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">I(X,Y)=H(X)-H(X|Y) \geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>&#xFF0C;&#x6240;&#x4EE5;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x2264;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X|Y)\leq H(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2264;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span>&#xFF0C;&#x7B49;&#x53F7;&#x5728;X&#x4E0E;Y&#x76F8;&#x4E92;&#x72EC;&#x7ACB;&#x65F6;&#x6210;&#x7ACB;&#x3002;<br>
&#x4ECE;&#x4FE1;&#x606F;&#x8BBA;&#x7684;&#x89D2;&#x5EA6;&#xFF0C;&#x6211;&#x4EEC;&#x53EF;&#x4EE5;&#x770B;&#x51FA;&#xFF1A;&#x4E24;&#x4E2A;&#x968F;&#x673A;&#x53D8;&#x91CF;&#x76F8;&#x4E92;&#x72EC;&#x7ACB;&#x7B49;&#x4EF7;&#x4E8E;&#x5B83;&#x4EEC;&#x4E4B;&#x95F4;&#x7684;&#x4E92;&#x4FE1;&#x606F;&#x4E3A;0.<br>
&#x8BE5;&#x7ED3;&#x8BBA;&#x8FD8;&#x53EF;&#x4EE5;&#x8FDB;&#x4E00;&#x6B65;&#x63A8;&#x5E7F;&#x5230;&#x4E09;&#x4E2A;&#x968F;&#x673A;&#x53D8;&#x91CF;&#x7684;&#x60C5;&#x51B5;&#x3002;<br>
&#x5BF9;&#x4E8E;&#x968F;&#x673A;&#x53D8;&#x91CF;X,Y,Z&#xFF0C;&#x6761;&#x4EF6;&#x71B5;H(X|Z)&#x662F;&#x7ED9;&#x5B9A;Z&#x65F6;X&#x5269;&#x4F59;&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#xFF0C;&#x5982;&#x679C;&#x518D;&#x8FDB;&#x4E00;&#x6B65;&#x7ED9;&#x5B9A;Y&#xFF0C;X&#x5269;&#x4F59;&#x7684;&#x4E0D;&#x786E;&#x5B9A;&#x6027;&#x53D8;&#x4E3A;H(X|Z,Y)&#x3002;&#x8FD9;&#x4E24;&#x8005;&#x4E4B;&#x5DEE;&#x5373;&#x4E3A;&#x7ED9;&#x5B9A;Z&#x65F6;&#x89C2;&#x6D4B;Y&#x7684;&#x53D6;&#x503C;&#x4F1A;&#x5E26;&#x6765;&#x7684;&#x5173;&#x4E8E;X&#x7684;&#x4FE1;&#x606F;&#x91CF;&#xFF0C;&#x5373;&#x7ED9;&#x5B9A;Z&#x65F6;X&#x548C;Y&#x4E4B;&#x95F4;&#x7684;&#x6761;&#x4EF6;&#x4E92;&#x4FE1;&#x606F;&#xFF0C;&#x5B9A;&#x4E49;&#x5982;&#x4E0B;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">;</mo><mi>Y</mi><mi mathvariant="normal">&#x2223;</mi><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Z</mi><mo stretchy="false">)</mo><mo>&#x2212;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Z</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(X;Y|Z)=H(X|Z)-H(X|Z,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">&#x2212;</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></span></p>
</section><section vertical="true" lineno="296" class="slide " data-line="296" data-h="7" data-v="2">
<p>&#x7C7B;&#x4F3C;&#x4E0A;&#x6587;&#x8BC1;&#x660E;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">;</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mi>Y</mi><mo separator="true">;</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(X;Y)=I(Y;X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span>&#xFF0C;&#x6211;&#x4EEC;&#x4E5F;&#x5BB9;&#x6613;&#x8BC1;&#x660E;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">;</mo><mi>Y</mi><mi mathvariant="normal">&#x2223;</mi><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mi>Y</mi><mo separator="true">;</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(X;Y|Z)=I(Y;X|Z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span></span></span></span></span><br>
&#x7C7B;&#x4F3C;&#x4E0A;&#x6587;&#x8BC1;&#x660E;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">;</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x2265;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">I(X;Y)\geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>&#x548C;<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>&#x2264;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X|Y)\leq H(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2264;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span>&#xFF0C;&#x6211;&#x4EEC;&#x4E5F;&#x5BB9;&#x6613;&#x8BC1;&#x660E;&#xFF1A;<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>I</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">;</mo><mi>Y</mi><mi mathvariant="normal">&#x2223;</mi><mi>Z</mi><mo stretchy="false">)</mo><mo>&#x2265;</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Y</mi><mo separator="true">,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>&#x2264;</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mi mathvariant="normal">&#x2223;</mi><mi>Z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
&amp; I(X;Y|Z)\geq 0 \\
&amp; H(X|Y,Z)\leq H(X|Z)
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000000000000004em;vertical-align:-1.2500000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.75em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2265;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&#x2264;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">&#x2223;</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">)</span></span></span></span><span class="vlist-s">&#x200B;</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span><br>
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